Optimization Applications of Compensators of Poisson Random Measures

Compensator or dual predictable projection is one of the main concepts of the general theory of stochastic processes, established by a French probabilistic school. In general, it is a considerably abstract concept and difficult to grasp. But in the Poisson case, it affords very powerful tools to calculate the expectation of certain stochastic integrals with respect to Poisson processes and random measures. Naturally, compensators can be used to solve certain problems of applied probability, involved in Poisson processes and random measures. In this direction, Brémaud and Jacod [1] had used the compensators of Poisson processes to solve the optimal dispatching problem, investigated by Ross [4] originally. In this note, we use the compensators of Poisson random measures to solve two optimal stopping problems: house selling and the burglary problem. The solutions are simple and completely rigorous as well. Our objective is to attract more attention to this method of using compensators. To facilitate reading, we introduce some fundamental results on marked point processes and Poisson random measures and their compensators.